Ref: See http://docs.drush.org/en/8.x/install/
Drush is completely independent of your Drupal installation(s).

To update it to a later version, follow the same instructions as initial installation, and simply place the new version of Drush in place of the old.

Windows XP / Windows 7

The article "How to manually upgrade to Drush 5.x and 6.x on Windows XP and 7" provides instructions for manually upgrading to the latest development version of Drush in Windows. The instructions have been tested on Windows XP and Windows 7.

Linux

Note: If you already have Drush installed, you can simply run drush dl drush for it to download itself.

Applying the destination you can install Drush in the location you desire. E.g. In most Linux distributions:

sudo drush dl drush --destination='/usr/share'

Note: Do not run the update command from the same directory where drush is installed or you will get an error (cannot create directory), e.g. do not run the above command from the /usr/share/drush directory. You CAN run it from the /usr/share directory (per the above example).

Suggestion: To avoid accidentally destroying your 'aliases.drushrc.php' if you are using one, put that file in '.drush' or '/etc./drush' and NOT in your drush directory.

Other solutions (mileage may vary):